Apparently you're supposed to plug in the numbers 1 through 9, using each digit once, in each of the blank spaces. Using the identified operations and the inserted values, a correct answer would lead to a solution of 66 (those double dots stand for division, by the way).

Supposedly there are quite a few solutions. A commenter on a Gizmodo article broke down the grid into an equation to help follow the order of operations (parenthesis, exponents, multiply, divide, add, subtract). They then went with an educated fill-in and guess:

There are other possible solutions, but the first one I came up with was: 6, 9, 3, 5, 2, 1, 7, 8, 4 in that order ... How I got to it was assigning each blank to a letter and writing the whole thing as an equation then grouping like terms.

So... a+13(b/c)+d+12e-f+(gh/i)-21=66.

Then you add 21 to both sides and a+d-f+13(b/c)+12e+(gh/i)=87. And since you’re limited to plugging in numbers between 1-9 for a-i, you see that there’s only so many combinations that will yield numbers close to 87. So then you start guessing and checking how large you have to make the various numbers to make it work. You need smaller numbers in the denominators and the subtraction and larger numbers being multiplied.

The Guardian's Alex Bellos has promised to upload a solution "later," but if anyone has an easier approach, leave it in the comments.

Phew.

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Math Art

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Albrecht Dürer

Melencolia I is a 1514 work by the German Renaissance artist Albrecht Dürer. In the top right hand of the oft-theorized print contains a magic square, in which the numbers in the four quadrants, corners and centers equal the same number, 34. (Which also happens to be in the Fibonacci sequence.)
Campbell Dodgson wrote of the work: "The literature on Melancholia is more extensive than on any other engraving by Dürer: that statement would probably remain true if the last two words were omitted."